Bifurcational instability of an atomic lattice

In a recent article N.H. Macmillan and A. Kelly (1972) have confirmed on the basis of a linear eigenvalue analysis that a mechanically stressed perfect crystal can exhibit a bifurcational instability at stresses ranging to 20 per cent below that of the limiting maximum of the primary stress-strain curve. The question thus arises as to whether the branching point is in a non-linear sense either stable or unstable. In the former case, perfect and slightly imperfect crystals would be capable of sustaining stresses over and above the eigenvalue critical stress. In the unstable case, however, this eigenvalue stress would represent the ultimate strength of a perfect solid, while an imperfect crystal would fail at a limiting stress substantially below the eigenvalue.At 20 per cent below the limit point such a branching point is essentially distinct, and the non-linear stability analysis needed to answer this question is provided by a recently established general branching theory for discrete conservative systems. Often, however, the two critical equilibrium states are much nearer than this, and the branching theory is here suitably extended to cover the case of near-compound instabilities.An illustrative study of a close-packed crystal under uniaxial tension is next presented. A kinematically-admissible displacement field is employed and a bifurcation point is located on the primary equilibrium path just before the limiting maximum, the eigenvector being associated with a transverse shearing strain. Under these conditions a corresponding small transverse shearing stress would represent an ‘imperfection’, and the non-linear branching problem is next studied using the new general theory. This shows (in excellent quantitative agreement with an ad hoc numerical solution) that the branching point is non-linearly unstable with a quite severe imperfection-sensitivity which manifests itself as a sharp cusp on the failure-stress locus.